Ugly Armature Mesh Deformation in Edit Mode
Okay!  I found the answer!



The problem was that I had a subdivision surface modifier on the mesh but I didn't apply it yet.  The armature was using the subdivision modifier to make it curve smoothly, but I was unable to edit those vertices that were making those weird curves because I didn't apply it to the mesh yet, all I was able to edit were the small amount of vertices that were there.  So I applied the modifier to my mesh and now it doesn't curve anymore because I have control of those vertices in edit mode.  



Unfortunately I had some shape keys to the mesh already and I had to delete them in order to apply the modifier.  Oh well!